Poisson and symplectic functions in Lie algebroid theory

TL;DR

This paper explores the use of Poisson and symplectic functions within Lie algebroid theory, revealing a correspondence between non-degenerate Poisson and symplectic functions and analyzing twisted structures.

Contribution

It demonstrates that various Lie algebroid brackets can be uniformly described using Poisson and pre-symplectic functions, establishing a correspondence and analyzing twisted structures.

Findings

Established a one-to-one correspondence between non-degenerate Poisson and symplectic functions.

Unified different Lie algebroid brackets under a common framework using Poisson and symplectic functions.

Analyzed the differential structures resulting from twisting Lie algebroids with background data.

Abstract

Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.